Proof Problems Without Stated Conclusions

When it comes to proof problems, the standard format is to say, "Prove that such-and-such holds." In other words, the conclusion, "such-and-such," is already written in the question. But isn't it boring to prove something you already know is true? By stating that it "holds," aren't you robbing the students of the fun? The scientific attitude is "hypothesis to verification." Even mathematicians stake their lives on "conjecture to proof." So, is it really okay for the question writer to just give away the conclusion?
From the student's perspective, being told the conclusion and then being asked to "go ahead and prove it" is neither a hypothesis nor a conjecture. It is an absolute truth. If it weren't, it would be a mistake in the question. Isn't it strange to be motivated to prove something when you don't have a shred of doubt? Isn't the necessity of a proof born precisely from not knowing whether something is true or not?
Thinking this way, I tried setting problems with the conclusion (or part of it) left blank. The idea is for students to first form a conjecture and then prove it themselves. Of course, the problems are designed so that the blank has a unique answer. When I do this, the correct answer rate drops significantly. In some cases, it falls apart completely.
However, by doing this, I can let students experience the most enjoyable part of mathematics: "forming hypotheses and conjectures." And if they can prove it, only then do they become convinced that it is true. I believe this is how proof problems should be.
I will present five problems below. All of them are from the scope of junior high and high school geometry.

Find the congruent triangles

This is a proof problem about "triangle congruence" learned in the first year of junior high school. It is a typical problem that is almost certainly found in every workbook. However, usually, the conclusion is stated clearly in the question. Above, I left that part blank so that students would first find the congruent triangles and then prove it.
Now, when it comes to finding congruent triangles, you have to draw auxiliary lines. By the way, when the conclusion is written, it is the same as having the auxiliary lines drawn in advance, so the trial-and-error process of drawing auxiliary lines is also lost.
In contrast, in this case, you would try drawing auxiliary lines here and there to first find triangles that look congruent. And even if you find them, that is just a conjecture, so the respondent will not be convinced immediately. They will likely try to verify it while thinking, "Is it really congruent?" I believe this movement of the heart is the joy of mathematics, don't you?

※ For the answer to 【1】, please click here.

Find the four points on the same circle

Next is a problem where you first find four points on the same circle and then prove it. Let's try to find them. First, does it look like "the four points B, P, C, and Q are on the same circle... (1)"? Unfortunately, those four points are not on the same circle.
If you look for others, does it look like "the four points A, B, C, and D are on the same circle... (2)"? Actually, that is the correct answer.
By the way, when I had students do this problem, a certain number of students put (1) in the blank. Of course, (1) does not hold, so it cannot be proven, but they write something that looks like a proof. And perhaps among those students who wrote such an answer, a certain number would write a correct proof if the problem were presented with the correct conclusion (2) written out. In other words, even if someone can write a correct proof for a problem where the conclusion is given, I doubt whether they really understand it properly.

※ For the answer to 【2】, please click here.

Are they congruent?

In this problem, it does not say "△ABE ≡ △DCE holds." The phrasing "Does it hold?" is the point. The purpose of this problem is: "If you say it holds, prove it. If you say it doesn't hold, provide a counterexample."
Well, what do you think? First, you can immediately see that the angles of the triangles are equal, so you know that △ABE and △DCE are similar. Therefore, if you can find that "the length of some side of the triangle is equal," you can say △ABE ≡ △DCE. Now, which side lengths can you say are equal? How can you say that?
I tried this problem in a second-year junior high school class. A student said, "I did it!" so I had them show me. I replied, "Does this part really hold? Isn't your reasoning a bit loose?" and sent them back. Then another student said, "I did it!" and another said, "This time it's definitely right!" and they kept coming to show me, so after a while, a long line formed.
I said, "You managed to prove it? That's a shame. You're all wrong. Because it 'doesn't hold.' I know a counterexample." Then the students started looking for a counterexample, but you can't find it by looking blindly. Rather, by carefully thinking, "Does this really hold? Are there any other possibilities?" with the intention of proving congruence, other possibilities—that is, counterexamples—become visible.
For example, a common mistake is to say "Since AC=BD, ∠ABC=∠DCB," but while "inscribed angles subtended by equal arcs are equal" holds, "inscribed angles subtended by equal chords are equal" does not. This is because even if you fix one chord, there are two inscribed angles. And if you think about this thoroughly, you will see a counterexample.
Alternatively, if you say "Let F be the intersection of AB and CD," you can somehow create something like a proof. However, if you think, "Do AB and CD even intersect in the first place?" you will see a counterexample.
There is also a way to use the "Power of a Point Theorem," but if you do it sloppily, it looks like a proof of congruence, but if you do it carefully, you will see a counterexample from there as well.
Once you see the counterexample, I'm sure you'll say, "I see." Please give it a try.

※ For the answer to 【3】, please click here.

If you draw three angle bisectors...

【4】 There is a △ABC that is not an isosceles triangle. Let D be the intersection of the internal angle bisector of ∠A and line BC, E be the intersection of the internal angle bisector of ∠B and line CA, and F be the intersection of the internal angle bisector of ∠C and line AB. At this time, ______________________________ .

This problem also hides the conclusion. The purpose of this problem is for the respondent to first answer the conclusion and then prove it.
Now, many people probably know the conclusion to this problem. The conclusion is "the three lines AD, BE, and CF intersect at one point." Many people also know that this point is called the "incenter."
So, how about the next problem?

　Problem 【4'】 is the same as Problem 【4】, except that the three "internal angles" have been changed to "external angles." That is the only difference, but many people likely do not know the conclusion in this case. And when that happens, you will first want to predict the conclusion, but since you won't be certain, you will likely want to prove it properly. This problem is the main topic.
　By the way, when I covered this in class, I had 【4】 printed on the handout, and just when the students thought, "This is a piece of cake," I said, "Oh, wait, I found a typo! It should be 'external angles,' not 'internal angles.' Please rewrite 'internal' to 'external' and solve it." I was acting poorly, but for the students, it was far from a piece of cake; it suddenly became a difficult problem.
　The reason is that it is quite hard to even predict the conclusion for this problem. If you draw the figure sloppily, it will be impossible. Since the three points D, E, and F are located quite far outside the triangle, the trick is to draw a small, obtuse triangle in the middle of the paper. Then, bisect the angles quite accurately and draw straight lines. If you do that, you might be able to make a prediction. But that is just a prediction, so it won't be convincing enough for the solver. To confirm that it holds for any triangle, you really have to prove it.
　By the way, the proofs for 【4】 and 【4'】 are almost the same, if you can say they are almost the same. There is a "formula for angle bisectors," and the formulas that hold for bisecting internal angles and external angles are very similar. First, you use that formula, and the subsequent development of the proofs for 【4】 and 【4'】 is also very similar. It is only at the very end that they are completely different. Comparing the two deepens one's understanding of plane geometry.

※ For the solution to 【4】, click here, and for the solution to 【4'】, click here.

The point where the sum of distances from the vertices of a quadrilateral is maximized

　First, please try to predict. I suspect many people would predict "the intersection of diagonals AC and BD... (3)." I predicted that too. Now, is that correct?
　In 《Figure 1》, if we let P be the intersection of diagonals AC and BD, and take P' at a position different from P, then "PA+PB+PC+PD < P'A+P'B+P'C+P'D" holds. The reason, in short, is that "it is shorter to go in a straight line than to take a detour." Thus, (3) can be proven in 《Figure 1》.
　But can (3) be said for any quadrilateral?

　What about the quadrilateral in 《Figure 2》? In this case, there seems to be a point where the value of PA+PB+PC+PD is even smaller than at the "intersection of diagonals AC and BD." Let's make a prediction here as well. In the case of 《Figure 2》, what would you predict?
　I suspect many people would predict "point C." I predicted that too. However, I cannot prove it. In 《Figure 1》, my prediction was correct. I was also able to prove it. However, in 《Figure 2》, I can make a prediction, but I cannot prove it. Therefore, I have no certainty. Or rather, the fact that I cannot prove it even after trying hard might mean my prediction is wrong.
　So, I don't know the answer to Problem 【5】. But isn't that fine? Since the title of this presentation is "Proof Problems Without Stated Conclusions," it might be acceptable to also have "Proof Problems Where the Conclusion Is Unknown" based on that theme. I really don't know, so if anyone has solved it, please let me know.

※ For the proof of 【5】 《Figure 1》, click here.

　If you state the conclusion, the solver will have no shred of doubt. It is absolutely correct. If it weren't, it would be a mistake in the problem. But isn't it boring to prove something you already know is true? Isn't that an unnatural motivation?
　If you state the conclusion, it becomes a task rather than a verification. And when that happens, you can write something that looks plausible even if you don't really understand it. Where did the joy of discovery and the surprise of finding it go?
　So, what should be done? It's simple. Just don't state the conclusion. That alone makes proof problems fun. Problem setters, please try your best not to state the conclusion.

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~ My Mathematics Education Theory ~
▷ Proof Problems Without Stated Conclusions + Solution Edition
▷ It's Becoming an Era Where Mathematicians Can Sell
▷ The Magic of "3"

~ Mathematics Problem Collection for Thinking ~
▷ Instant kill if you understand, self-evident if you see the answer
▷ Pythagorean triples consist of multiples of 3, 4, 5
▷ Similar figures
▷ Let's think using the "Pigeonhole Principle"
▷ Proof Problems Without Stated Conclusions